i. Let R be a set with 2 laws of composition satisfying all ring axioms except the comutative law for addition. Use the distributive law to prove that the commutative law for addition holds, such that R is a ring.
ii. Find generator for the kernel of the map Z[x]->C defined by x->sqrt(2)+sqrt(3).
Proof: We consider any two elements , we know that satisfies all axioms of a ring except the commutative law for addition, so we want to show that .
First, according to the distributive law, we have , where is ...
The commutative law for addition are examined for the ring and kernel.